Question

A circle is inscribed in a triangle with sides $3,4$ and $5cm.$ The radius of the circle is ___ $cm.$

Answer 1

Join the centre of the circle and the vertices of the triangle. Observe that the sides of the triangle become the tangents to the circle. Hence, the radii of the circle as shown in the question become the heights of the smaller triangles.

Now, area of $△$ABC $=$ area of $△$OCB $+$ area of $△$OBA $+$ area of $△$OCA
$\Rightarrow (\frac{1}{2}\times r\times 4)+(\frac{1}{2}\times r\times 3)+(\frac{1}{2}\times r\times 5)=\frac{1}{2}\times 4\times 3$
$\Rightarrow \frac{1}{2}\times r\times (3+4+5)=\frac{1}{2}\times 3\times 4$

$\Rightarrow 6r=6$
$\Rightarrow r=1cm$